Figure 3.1.: Muon neutrino survival probability using Eq. (3.2.11) as a function of energy. The blue curve shows the oscillation probability without sterile mixing, while the magenta solid (dashed) curve shows the probability for |Uµ4|2 = 10−1.6,
|Uτ 4|2 = 0.15, and δ24= 0 (δ24= π). The baseline has been set to the diameter
of the Earth.
it can change the sign of the interference term between the atmospheric and the sterile terms in the expression for the energy and the mixing angle in matter. As an example of the impact of the phase, in Figure 3.1 the muon neutrino survival probability as a function of the energy for |Uµ4|2 = 10−1.6, |Uτ 4|2 = 0.15, L = 1.2× 104 km, and two
different values of the phase, δ24 = 0 (solid line) and δ24 = π (dashed line) is shown.
As comparison, the muon neutrino disappearance oscillation probability for zero sterile mixing is also shown. These values of the sterile matrix elements are at the border of the 90% C.L. region of SuperKamiokande. The sign of the interference term can also be changed by changing the mass ordering (i.e., the sign of ∆m2
31) or by switching between
neutrinos and antineutrinos (i.e., changing the sign of VNC). However, neither IceCube
nor SuperKamiokande or DeepCore can distinguish between neutrinos and antineutrinos so this dependence is diluted in their data.
Conversely, experiments such as CHORUS and NOMAD explored the same parameter space but instead exploiting the νµto ντ appearance channel with negligible matter effects
leading to Pµτ ' 4|Uτ 4|2|Uµ4|2sin2 ∆m2 41L 4E . (3.2.16)
3.3. Simulation and results
One year of high-energy through-going muons released by the IceCube collaboration [110] for the last IceCube detector stage with 86 strings will be analyzed. The data sample con- sists of up-going track events so as to avoid the background from cosmic ray muons giving, after all cuts, a sample purity better than 99.9%. Hence, the distances the signal neutrinos travel are of the order of 104 km. The selected events have reconstructed energies between
400 GeV and 20 TeV and cosine of the reconstructed zenith angle between −1 and 0.2. The sensitivity that a full 8-year IceCube sample would have as well as the prospects for an exposure equivalent to 20 years of IceCube data will also be forecasted. For our simula- tions, the neutrino flux computed with the analytic air shower code [217] using the cosmic
results do not change significantly under the assumption of different fluxes, such as using the cosmic ray flux from the poly-gonato model [220,221] or the Zatsepin-Sokolskaya [222] model updated with measurements by PAMELA [223] together with the hadronic model SIBYLL2.3, RC1, point-like [224] or QGSJET II-04.
The propagation of the neutrinos was simulated using the nuSQuIDS software [215,216], where the PREM profile [225] is implemented for the Earth matter density. Since we are interested in the averaged out regime our simulations were performed with a sterile mass squared difference of ∆m2
41= 103 eV2, but we have verified that changing this parameter
does not alter the results as long as ∆m2
41& 100 eV2 as expected.
Since neutrino and antineutrino interactions cannot be distinguished on an event basis, the signal will contain both νµ and ¯νµ events. After propagating the flux for every value
of the sterile neutrino parameter, the Monte Carlo provided with the data releas [110] has been used to compute the expected number of events Nth,i in every bin of reconstructed
zenith angle.
In order to obtain the expected significance of the bounds on the sterile mixing param- eters, we adopt a Poisson log-likelihood given by
L =−X
i
Nth,i− Nd,i+ Nd,ilog
Nd,i
Nth,i
, (3.3.1)
where the Nth,i and Nd,i are the predicted and observed number of events given a set of
parameters in bin i, respectively, and the sum is taken over all the reconstructed zenith angle bins i.
The log-likelihood has been maximized for a number of nuisance parameters to include the effect of possible systematic errors. In particular, the uncertainty in the pion-kaon ratio of the initial flux (π/k), the efficiency of the digital optical modules (DOMs), and the overall flux normalization have been considered. Since the observable is energy independent for large values of the sterile neutrino mass (see Eq. (3.2.15)), only one energy bin has been considered and the uncertainty in the energy spectrum slope has been neglected, while 40 bins for the reconstructed zenith angle have been adopted. For the pion-kaon ratio a Gaussian prior with σπ/k = 0.05 has been adopted and no prior for the DOM efficiency
or the overall flux normalization has been assumed. The standard oscillation parameters used in the simulations were set to their respective best-fit values from Tab. 3.1. To find the confidence regions from the log-likelihood differences we assume that the prerequisites for Wilks’ theorem [226] holds so that likelihood ratios can be directly converted to a confidence level.
In the left panel of Figure 3.2, the 90% C.L. constraints (for 2 degrees of freedom) obtained for the public 1-year data (pink contours) in the|Uµ4|2-|Uτ 4|2-plane is presented.
The existing bounds from SuperKamiokande [192] and DeepCore [206] at the same C.L. are also shown for comparison by the hatched gray area. At 90% C.L. present data prefer some degree of sterile mixing and we find that zero sterile mixing is disfavoured at 2.3σ (1 degree of freedom3). The preference for non-zero sterile mixing is independent on
the atmospheric sterile neutrino flux adopted in the analysis but its significance varies between 1.6 and 3.0 σ with the different models tested. Given this preference for non-zero sterile mixing, the current constraints from IceCube do not improve upon the combined bounds from SuperKamiokande and DeepCore at 90% C.L. In the right panel, the same information is shown at 99% C.L. In this case, the present 1-year data gives an upper bound that already slightly improves upon the present SuperKamiokande and DeepCore constraints, ruling out the white region in the plot. We have also checked how these results
3Note that if U
3.3. Simulation and results 39 ★ ★ ��+�� ���������� ����� ���������� �� (� ���� ����) �� (� ����� ��������) 100-3 10-2 10-1 0.05 0.1 0.15 0.2 0.25 Uμ42 U τ4 2 ★ ★ ��+�� ���������� ����� ���������� �� (� ���� ����) �� (� ����� ��������) 100-3 10-2 10-1 0.05 0.1 0.15 0.2 0.25 Uμ42 U τ4 2
Figure 3.2.: The left (right) panel shows in pink the constraints at 90% (99%) C.L. for the sterile mixing elements from the released 1-year data. The cyan region shows at the same C.L. the forecast for 8 years of IceCube data assum- ing as true values |Uµ4|2 = 10−2, |Uτ 4|2 = 0.1, δ24 = 0 (marked with a
star). The full (dashed) lines show the bounds for δ24 = 0 (δ24 = π). The
solid (dashed) hatched regions are disfavoured by SuperKamiokande [192] and DeepCore [206] (NOMAD [208]) data at the same C.L.
depend on the binning in energy and zenith angle. We have seen that when the data is also binned in energy, the case of no active-sterile mixing becomes slightly less disfavoured. Using different combinations of fluxes and binning, the no-mixing scenario is disfavoured at between 0.74 and 3.1 σ, depending on the combination. In particular, for the case of 10 energy bins and 21 bins in the zenith angle, as presented in ref. [110], the significance varies between 0.75 and 3.0 σ.4
The physics reach of an 8-year run of IceCube data if the present preference for sterile mixing is maintained is also shown in cyan. In particular, the present best-fit value of |Uµ4|2 = 10−2, |Uτ 4|2 = 0.29 lies in the already disfavoured region by DeepCore and
SuperKamiokande. Due to the hyperbola-shaped degeneracy of the oscillation probability in the|Uµ4|2-|Uτ 4|2-plane, there are values of the sterile oscillation parameters that provide
an almost equally good fit without being in tension with the other νµdisappearance present
data. Remarkably, theses values of Uτ 4 are also compatible with the sterile neutrino
interpretation [202] of the upward directed cosmic ray shower observed by ANITA [203]. Indeed, the sterile neutrino interpretation of the ANITA results requires that the sterile neutrino mass is between∼ 102 and ∼ 106 eV, which would also fall in the averaged out
regime for IceCube studied here. However, all the parameter space preferred by IceCube at the 90% C.L. is disfavoured by NOMAD [208] with the same significance. Indeed, the null results in their ντ search translates through Eq. (3.2.16) into|Uµ4|2|Uτ 4|2< 8.3· 10−5
at the 90% C.L. for ∆m2
41 & 100 eV2. Nevertheless, the channel and underlying physics
explored to obtain the bounds are very different in the two sets of experiments. While
4
After this work was submitted to the arXiv, a different analysis [101] did not find any preference for non- zero active-sterile mixing. In particular ref. [101] adopts the same binning as ref. [110] and marginalizes over fluxes. Since in this case we find that no sterile mixing is disfavoured by only ∆χ2 = 0.56, the different approaches to the treatment of systematic errors can easily account for the small discrepancy between the two results.
������� (� ����� ��������) ������� (�� ����� ��������) 100-3 10-2 10-1 0.05 0.1 0.15 0.2 Uμ42 U τ4 2 ������� (� ����� ��������) ������� (�� ����� ��������) 100-3 10-2 10-1 0.05 0.1 0.15 0.2 Uμ42 U τ4 2
Figure 3.3.: The left (right) panel shows the expected constraints in absence of sterile neu- trino mixing at 90% (99%) C.L. for the sterile mixing elements from datasets composed of 8-year (cyan) or 20-year (purple) of IceCube data. The full (dashed) curves show the bounds for δ24 = 0 (δ24 = π). The solid (dashed)
gray hatched regions are disfavoured by SuperKamiokande [192] and Deep- Core [206] (NOMAD [208]) data at the same C.L.
SuperKamiokande, DeepCore and IceCube analyze νµ disappearance and the steriles are
probed via their matter effects as shown in Eq. (3.2.15), NOMAD and CHORUS searched for ντ appearance essentially in vacuum through Eq. (3.2.16). Thus, in presence of non-
standard matter effects (also conceivably in the sterile sector) the two results could still be reconciled if a stronger tension should remain upon including more IceCube data. We therefore simulate 8 years of IceCube data assuming |Uµ4|2 = 10−2, |Uτ 4|2 = 0.1, and
δ24 = 0 as the true oscillation parameters. As can be seen in Figure 3.2, the expected
confidence region region shrinks significantly with the additional statistics, while keeping its shape. In particular, if the values of the sterile neutrino mixing marked by the star were realized in nature, 8-years of IceCube data would disfavour no sterile mixing around the 5σ level.
The capability of larger IceCube samples to improve the present constraints on sterile mixing in absence of sterile neutrinos have been also studied. In Figure 3.3, the contours for 90% (left panel) and 99% C.L. (right panel) expected exclusion limits in the |Uµ4|2-
|Uτ 4|2-plane together with the existing bounds from SuperKamiokande and DeepCore are
presented. The bound on|Uµ4|2 from 8 years of IceCube would improve over present con-
straints between a factor 1.3 for vanishing values of|Uτ 4|2 to around an order of magnitude
for|Uτ 4|2 close to 0.1. Similarly, for|Uµ4|2 ∼ 10−2, the constraint on|Uτ 4|2 would improve
around a factor 5. In particular, the present best fit for non-zero sterile mixing would be excluded at high significance (more than 5σ) and most of the currently preferred parameter space at 90% C.L. (pink area in the left panel of Figure 3.2) disfavoured. Comparatively, increasing the statistics up to 20-year of IceCube data yields a more modest improvement in sensitivity. Remarkably, not even the 20-year scenario would improve over the present NOMAD limit of|Uµ4|2|Uτ 4|2 < 8.3· 10−5 at the 90% C.L. Nevertheless, we consider the
two constraints complementary given the different physics probed by each of them. The effect of the CP-violating phase δ24 is also shown. In particular, the solid lines